#8202396

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,11)(4,13)(5,10)(6,8)(7,14)(9,15)(12,16)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32)(33,34,40,44)(35,45,39,41)(36,47,46,42)(37,48,43,38);(1,3,6,14)(2,7,8,11)(4,10,15,12)(5,13,16,9)(19,20)(21,30)(23,26)(24,29)(33,35,40,39)(34,41,44,45)(36,43,46,37)(38,47,48,42);(2,8)(5,16)(7,11)(9,13)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36)(34,42)(35,37)(38,45)(39,43)(40,46)(41,48)(44,47);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,7,16,11)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,37,40,43)(34,38,44,48)(35,46,39,36)(41,42,45,47);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,38)(34,43)(35,47)(36,45)(37,44)(39,42)(40,48)(41,46);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,7,15,11)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,31)(29,30)(33,39,40,35)(34,41,44,45)(36,43,46,37)(38,42,48,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8109592);